Traditionally, streamlines in discrete models of natural oil reservoirs are traced by solving an ordinary differential equation in a cell-by-cell way, using analytical or numerical solutions, considering the local velocity of each cell. This strategy has a disadvantage: the streamline is traced considering a discrete, and so discontinuous, vector field. Furthermore, for massive models, to solve the equation in a cell-by-cell way may be inefficient. In this work, we explore a different strategy: the streamline tracing considers a continuous vector field represented by the discrete model. Therefore, we propose: (i) to use a spatial structure to speed up the point location process inside the reservoir model; (ii) to use spherical interpolation to obtain the velocity field from the discrete model; (iii) to use an adaptive numerical method to control the numerical error from the integration process. The results obtained for actual reservoir models demonstrate that the proposed method fulfills the precision requirements, keeping a good performance.